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3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 34 Vol. 9 N.º 2 Junio - Septiembre

49

ALGORITHMIC EFFICIENCY INDICATOR FOR THE OPTIMIZATION OF ROUTE SIZE

Ciro RodríguezNational University Mayor de San Marcos. Lima, (Perú).

E-mail: [email protected] ORCID: https://orcid.org/0000-0003-2112-1349

Miguel SifuentesNational University Mayor de San Marcos. Lima, (Perú).


Freddy KasengNational University Federico Villarreal. Lima, (Perú).


Pedro LezamaNational University Federico Villarreal. Lima, (Perú).


Recepción: 06/02/2020 Aceptación: 23/03/2020 Publicación: 15/06/2020

Citación sugerida:Rodríguez, C., Sifuentes, M., Kaseng, F., y Lezama, P. (2020). Algorithmic efficiency indicator for the optimization of route size. 3C Tecnología. Glosas de innovación aplicadas a la pyme, 9(2), 49-69. http://doi.org/10.17993/3ctecno/2020.v9n2e34.49-69

mailto:[email protected]

https://orcid.org/0000-0003-2112-1349?lang=en



https://orcid.org/0000-0002-2878-9053


https://orcid.org/0000-0003-2112-1349?lang=en

http://doi.org/10.17993/3ctecno/2020.v9n2e34.49-69



50 http://doi.org/10.17993/3ctecno/2020.v9n2e34.49-69

ABSTRACTIn software development, sometimes experienced programmers have difficulty determining before

performing their tests, which algorithm will work best to solve a particular problem, and the answer

to the question will always depend on the type of problem and nature of the data to be used, in this

situation, it is necessary to identify which indicator is the most useful for a specific type of problem

and especially in route optimization. Currently, there are techniques and algorithms used in Artificial

Intelligence, which, however, cannot display their potential without a well-defined data set. The paper

seeks to explain and propose an algorithm selection indicator to build a consistent data set, given the lack

of availability of data that allows the best route size decision to be made.

KEYWORDSSoftware development, Algorithm efficiency, Indicator, Route size.




1. INTRODUCTIONThe type of problem and nature of the data to be used is necessary to identify which indicator is the

most useful for a specific kind of problem and especially in route optimization. As Microsoft (n.d.)

explain are several ways to make the selection. In Machine Learning, cross-validation is one of the most

commonly used. It was finding the best set of parameters like space, cross-validation, metric, accuracy. It

is important to formulate, evaluate, and compare the values of the parameters, after the evaluation and

comparison, it is possible to choose the best alternative as shown in Figure 1.

How to select machine learning algorithms

What do you want to do with your

data?Algorithm Cheat Sheet

Additional requirements Accuracy Training

time Linearity Number of parameters

Number of features

Figure 1. What are the requirements of your data science scenario? Source: (Microsoft, n.d.).

In the search for the most appropriate algorithm to implement in the route optimizer we will have three

candidates for the best indicator: the algorithmic efficiency, the size (sum of weights of the edges) of the

route obtained and the following relation, derived from the previous indicators. The witness search is

expensive (Dibbelt, 2016), and therefore the witness search is usually aborted after a certain number of

steps. If no witness was found, is assumed that none exists and add a shortcut, as shown in Figure 2.

x

v y

z

1

11

2

Figure 2. Contraction of v. Source: (Dibbelt, 2016).




Figure 2 sows pair (x, y); if is considered first, a shortcut {x, y} with weight 3 is inserted. If the pair (x, z)

is considered first, an edge {x, z} with weight 2 is inserted. This shortcut is part of a witness x → z → y

for the pair x, y. The shortcut {x, y} is thus not added if the pair x, z is considered first.

0

1

2

1

5

20

Figure 3. The minimum path from node 0 to node 2.

Figure 3 shows the result of using the Dijkstra algorithm in the graph, which presents the minimum sum

of edges to get from node 0 to node 2, however, it is not the most efficient of the algorithms presented

in this research paper.

2. CONCEPTUAL FRAMEWORK

2.1. ARTIFICIAL INTELLIGENCE AND DATASETS

Forecasts say that in the next decade, there will be approximately more than 150,000 million sensors

connected to the network (more than 20 times the Earth’s population). These data help AI devices

to think as we think, accelerating their learning curve and automation of data analysis with all the

processed information; considering as much data the system receives, more learning and accuracy

becomes (PowerData, 2017).

Today, artificial intelligence can learn without human support. New cases are known daily, such as the

example of a Google DeepMind algorithm, which recently learned on its own how to win 49 Atari

games, without the need for any interaction from anyone.




In the past, AI growth was minimal for two main reasons: limited data sets that used representative data

samples, instead of using real-time data, and inability to analyze massive amounts of data in seconds

(PowerData, 2017).

2.2. THE LACK OF DATASETS

The ability to manage large volumes and data sources is enabling the capabilities of AI (Artificial

Intelligence) and machine learning . However, some organizations cannot yet exploit AI capabilities for

various reasons:

• The lack of data availability.

• Limited sample sizes negatively affect their abilities.

• The lack of appropriate technologies to analyze massive data in milliseconds.

2.3. ALGORITHMICS

Knowing what problem is going to be solved is only half the job. In dealing with problems, they generally

do not have a precise and straightforward specification of them. Problems such as creating a gourmet-

worthy recipe or preserving world peace may be impossible to formulate in a way that supports a

computer solution; Although it is believed that the problem can be solved on a computer, it is usual that

the distance between several of its parameters is at least considerable (Aho, Hopcroft, & Jefrey, 1988).

Often only through experimentation, is it possible to find reasonable values for these parameters. If it is

possible to express certain aspects of a problem with a formal model, it is generally beneficial to do so,

because once the problem is formalized, solutions can be sought based on a precise model and determine

if there is already a program that solves that problem. The algorithm has a remarkable importance in the

development of software when considering various algorithms for a given problem.




The algorithm is defined as the study of algorithms, which are a sequence of ordered and finite steps,

each of which has a precise meaning and can be executed with a limited amount of effort in a finite

time to solve a problem. By steps means the set of actions or operations that are performed on certain

objects. For the solution of a problem, a collection of algorithms is taken into account, depending on the

particular characteristics of the problem (selected algorithm) (“Complejidad algorítmica”, n.d.).

Input OutputAlgorithm

Figure 4. Determinism as a feature of the algorithms.

An algorithm as Figure 4 must have the following characteristics:

• Accuracy: An algorithm must be expressed without ambiguity.

• Determinism: Every algorithm must respond in the same way before the same conditions

• Finite: The description of an algorithm must be limited.

An algorithm must also achieve the following objectives, which are often contradicted:

• Simple: Make it easy to understand, code, and debug.

• Efficient: Efficient use of computer resources in time, space (memory) and processor, so that it

runs as quickly as possible.

It is usually difficult to find an algorithm that meets both, so a compromise must be reached that best fits

the requirements of the problem (“Complejidad algorítmica”, n.d.).




The algorithm allows evaluating the results of different external factors on the available algorithms in

such a way that it will enable to select the one that best suits the particular conditions. It allows indicating

how to design a new algorithm for a specific task.

It has been said that each instruction in an algorithm must have a “precise meaning” and must be

executable with a “finite amount of effort”; But what is clear to one person may not be clear to another,

and it is often challenging to demonstrate rigorously that an instruction can be done in a finite time.

Sometimes, it is difficult to show the sequence of instructions ends with some input, even if the meaning

of each instruction is very clear. However, an agreement is usually reached as to whether or not a

sequence of instructions constitutes an algorithm (Aho et al., 1988).

2.4. ALGORITHMIC COMPLEXITY

Algorithmic complexity represents the number of resources (temporary) that an algorithm needs to

solve a problem and therefore allows to determine the efficiency of the said algorithm (Universidad de

Valladolid Campus de Segovia). It is not referred to the difficulty in designing algorithms. The criteria

that will be used to assess algorithmic complexity do not provide absolute measures but measures relative

to the size of the problem. What is relevant, at first, is the measure of temporal complexity in terms of

the input data.

When a problem is resolved, there is often a need to choose between several algorithms. How should you

choose? Two objectives are often contradicted:

1. That the algorithm is easy to understand, code, and debug.

2. That the algorithm efficiently uses the computer resources and, in particular, that it be executed as

quickly as possible (“Complejidad algorítmica”, n.d.).




When writing a program to be used once or a few times, the first objective is the most important. In

this case, it is very likely that the cost of programming time dramatically exceeds the value of program

execution, so that the price to be optimized is that of writing the program (Aho et al., 1988).

On the other hand, when there is a problem whose solution is going to be used many times, the cost of

executing the program can significantly exceed that of writing, especially if large entries are given in

most executions. Then, it is more advantageous, from the economic point of view, to perform a complex

algorithm provided that the execution time of the resulting program is significantly shorter than that of a

more obvious program. And even in situations like that, it may be convenient first to implement a simple

algorithm to determine the real benefit that would be obtained by writing a more complicated program.

In the construction of a complex system, it is often desirable to implement a simple prototype in which

simulations and measurements can be made before engaging in the final design. This concludes that a

programmer must not only be aware of ways to get a program to run quickly but also know when to

apply these techniques and when to ignore them (Aho et al., 1988).

An algorithm will be more efficient compared to another, provided it consumes fewer resources, such as

the time and memory space needed to execute it.

The evaluation of the algorithms for efficiency will be taken into account:

• The growth rate in time. Runtime: It has to do with the time it takes for a program to run

(processing time).

• The growth rate in space.

Memory space: Study the amount of space that is necessary for operations during program execution

(storage and processing space).

The efficiency of an algorithm can be quantified with the following complexity measures:

• Temporary Complexity or Execution Time: Computation time necessary to execute a program.




• Spatial Complexity: Memory that an application uses for its execution. The memory efficiency

of an algorithm indicates the amount of space required to execute the algorithm; that is to say,

the area in memory that occupies all the own variables to the algorithm. To calculate the static

memory of an algorithm, the memory occupied by the variables declared in the algorithm is

added. In the case of dynamic memory, the calculation is not so simple since this depends on each

execution of the algorithm (“Complejidad algorítmica”, n.d.).

2.5. GRAPHS

A graph G is a set of points in space, some of which are linked by lines (Menendez, 1998), formally, a

graph G consists of two finite sets N, and A. N is the set of elements of the graph, also called vertices

or nodes. A is the set of arcs, which are the connections that are responsible for relating the nodes to

form the graph. Arcs are also called edges or lines. Nodes are often used to represent objects and arcs

to represent the relationship between them. For example, nodes can represent cities and arches the

existence of roads that communicate them. Each arc is defined by a pair of elements n1, n2 N to which

it connects. Although the parts are usually different, we will allow them to be the same node (n1 = n2).

Trees have been considered as a generalization of the list concept because they allow an element to have

more than one successor. Graphs appear as an extension of the tree concept since, in this new type of

structure, each element can have, in addition to more than one successor, several predecessor elements.

This property makes graphs the most appropriate structures to represent situations where the relationship

between the elements is completely arbitrary, such as road maps, telecommunications systems, printed

circuits, or computer networks. Although there are more complex structures than graphs, we will not

see them in this course. Graphs can be classified into different types depending on how the relationship

between the elements is defined: we can find directed or non-directed and labeled or unlabeled graphs.

We will use the tags when we work with the Dijkstra and Prim algorithms.




The graph is not directed if the arcs are formed by pairs of unordered nodes, not pointed; A graph is

directed, also called a digraph, if the pairs of nodes that form the arcs are ordered; they are represented

with an arrow indicating the direction of the relation u → v, these being a pair of nodes (Joyanes &

Zahonero, 2008).

In some instances it is necessary to associate information to the arcs of the graph. This can be achieved

through a label containing any useful information related to the arc, such as the name, weight, cost or a

value of any given data type. In this case we talk about labeled graphs. This label could mean the time

it takes for the flight between two cities or indicates what the input and output parameters are in the call

to a subprogram. An unlabeled graph is a graph where the arcs have no labels. In the case of the graph

that represents the traffic direction, the arcs can be labeled.

3. METHODOLOGY

3.1. SHORTEST ROUTE

Researchers consider the Shortest Route Problem as a central problem within the area of networks

due to the variety of practical applications, the existence of efficient solution methods, and the use of

subroutines in the search for the right solution in complex problems. However, in the subjects in which

these kinds of problems are addressed, they are usually presented in a simplified and unclear way, that

is, the importance of studying these types of problems is often not recognized or stressed. Theoretical

aspect and due to the great diversity of applications (Obregon, 2005), so this problem can be solved by,

among others, the Dijkstra and Prim algorithms.




3.2. MINIMUM PATH ALGORITHMS

A. Naive Algorithm

The naive algorithm for directed graphs has reasonably good efficiency; however, it cannot see beyond

the node following the current one of the network.

It is an algorithm that starts from a source node and goes exploring the paths to the rest of the nodes.

The following pseudocode shows its operation, using the priority queue as an auxiliary data structure

(Jungnickel, 1999).

7

The graph is not directed if the arcs are formed by pairs of unordered nodes, not pointed; A graph is directed, also called a digraph, if the pairs of nodes that form the arcs are ordered; they are represented with an arrow indicating the direction of the relation u → v, these being a pair of nodes (Joyanes & Zahonero, 2008).

In some instances it is necessary to associate information to the arcs of the graph. This can be achieved through a label containing any useful information related to the arc, such as the name, weight, cost or a value of any given data type. In this case we talk about labeled graphs. This label could mean the time it takes for the flight between two cities or indicates what the input and output parameters are in the call to a subprogram. An unlabeled graph is a graph where the arcs have no labels. In the case of the graph that represents the traffic direction, the arcs can be labeled. 3. Methodology 3.1. Shortest route Researchers consider the Shortest Route Problem as a central problem within the area of networks due to the variety of practical applications, the existence of efficient solution methods, and the use of subroutines in the search for the right solution in complex problems. However, in the subjects in which these kinds of problems are addressed, they are usually presented in a simplified and unclear way, that is, the importance of studying these types of problems is often not recognized or stressed. Theoretical aspect and due to the great diversity of applications (Obregon, 2005), so this problem can be solved by, among others, the Dijkstra and Prim algorithms. 3.2 Minimum path algorithms

A. Naive Algorithm

The naive algorithm for directed graphs has reasonably good efficiency; however, it cannot see beyond the node following the current one of the network.

It is an algorithm that starts from a source node and goes exploring the paths to the rest of the nodes. The following pseudocode shows its operation, using the priority queue as an auxiliary data structure (Jungnickel, 1999).

𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁(𝑔𝑔𝑔𝑔𝑁𝑁𝑔𝑔ℎ𝐺𝐺, 𝑠𝑠𝑠𝑠𝑠𝑠𝑔𝑔𝑠𝑠𝑁𝑁𝑁𝑁𝑠𝑠)𝒇𝒇𝒇𝒇𝒇𝒇𝑠𝑠 ∈ 𝑉𝑉[𝐺𝐺]𝒅𝒅𝒇𝒇𝑑𝑑𝑁𝑁𝑠𝑠𝑑𝑑𝑁𝑁𝑑𝑑𝑠𝑠𝑁𝑁[𝑠𝑠] = 𝐼𝐼𝑁𝑁𝐼𝐼𝐼𝐼𝑁𝑁𝐼𝐼𝐼𝐼𝐼𝐼𝑔𝑔𝑁𝑁𝑔𝑔𝑁𝑁𝑑𝑑𝑑𝑑𝑁𝑁𝑠𝑠𝑑𝑑𝑁𝑁[𝑠𝑠] = 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑠𝑠𝑁𝑁𝑁𝑁𝑑𝑑[𝑠𝑠] = 𝑓𝑓𝑁𝑁𝑓𝑓𝑠𝑠𝑁𝑁

𝑑𝑑𝑁𝑁𝑠𝑠𝑑𝑑𝑁𝑁𝑑𝑑𝑠𝑠𝑁𝑁[𝑠𝑠] = 0𝑁𝑁𝑑𝑑𝑠𝑠𝑁𝑁𝑔𝑔𝑑𝑑(𝑞𝑞𝑠𝑠𝑁𝑁𝑠𝑠𝑁𝑁, (𝑠𝑠, 𝑑𝑑𝑁𝑁𝑠𝑠𝑑𝑑𝑁𝑁𝑑𝑑𝑠𝑠𝑁𝑁𝑁𝑁[𝑠𝑠]))𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘! 𝑁𝑁𝑠𝑠𝑖𝑖𝑖𝑖𝑔𝑔𝑑𝑑𝑖𝑖(𝑞𝑞𝑠𝑠𝑁𝑁𝑠𝑠𝑁𝑁)𝒅𝒅𝒇𝒇𝑠𝑠 = 𝑁𝑁𝑒𝑒𝑑𝑑𝑔𝑔𝑁𝑁𝑠𝑠𝑑𝑑𝑒𝑒í𝑑𝑑𝑁𝑁𝑖𝑖𝑠𝑠𝑖𝑖(𝑞𝑞𝑠𝑠𝑁𝑁𝑠𝑠𝑁𝑁)𝑠𝑠𝑁𝑁𝑁𝑁𝑑𝑑[𝑠𝑠] = 𝑑𝑑𝑔𝑔𝑠𝑠𝑁𝑁 𝑁𝑁𝑑𝑑𝑠𝑠𝑁𝑁𝑔𝑔𝑑𝑑(𝑞𝑞𝑠𝑠𝑁𝑁𝑠𝑠𝑁𝑁, (𝑁𝑁, 𝑑𝑑𝑁𝑁𝑠𝑠𝑑𝑑𝑁𝑁𝑑𝑑𝑠𝑠𝑁𝑁[𝑁𝑁]))

B. Dijkstra’s algorithm

The Dijkstra algorithm, also called the minimum path algorithm, is a model that is classified within the

search algorithms. Its objective is to determine the shortest route, from the origin node to any node in the

network. Its methodology is based on iterations, so that, in practice, its development becomes difficult as

the size of the network increases, leaving it at a clear disadvantage, compared to optimization methods

based on mathematical programming (Jungnickel, 1999).




8

B. Dijkstra's algorithm

The Dijkstra algorithm, also called the minimum path algorithm, is a model that is classified within the search algorithms. Its objective is to determine the shortest route, from the origin node to any node in the network. Its methodology is based on iterations, so that, in practice, its development becomes difficult as the size of the network increases, leaving it at a clear disadvantage, compared to optimization methods based on mathematical programming (Jungnickel, 1999).

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷(𝑔𝑔𝐷𝐷𝐷𝐷𝑔𝑔ℎ𝐺𝐺, 𝐷𝐷𝑠𝑠𝑠𝑠𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝐷𝐷)𝒇𝒇𝒇𝒇𝒇𝒇𝑠𝑠 ∈ 𝑉𝑉[𝐺𝐺]𝒅𝒅𝒇𝒇𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] = 𝐼𝐼𝑠𝑠𝐼𝐼𝐼𝐼𝑠𝑠𝐼𝐼𝐼𝐼𝐼𝐼𝑔𝑔𝐷𝐷𝐷𝐷𝑠𝑠𝑑𝑑𝐷𝐷_𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠[𝑠𝑠] = 𝑠𝑠𝑁𝑁𝑁𝑁𝑁𝑁𝐷𝐷𝑠𝑠𝑠𝑠𝑑𝑑[𝑠𝑠] = 𝑓𝑓𝐷𝐷𝑓𝑓𝐷𝐷𝑠𝑠

𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝐷𝐷] = 0𝐷𝐷𝑑𝑑𝐷𝐷𝑠𝑠𝐷𝐷𝐷𝐷(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠, (𝐷𝐷, 𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝐷𝐷]))𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘! 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑖𝑖𝑖𝑖𝑔𝑔𝐷𝐷𝑖𝑖(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝒅𝒅𝒇𝒇𝑠𝑠 = 𝑠𝑠𝑒𝑒𝐷𝐷𝐷𝐷𝐷𝐷𝑠𝑠𝐷𝐷_𝑖𝑖í𝑑𝑑𝐷𝐷𝑖𝑖𝑠𝑠𝑖𝑖(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝐷𝐷𝑠𝑠𝑑𝑑[𝑠𝑠] = 𝐷𝐷𝐷𝐷𝑠𝑠𝑠𝑠𝒇𝒇𝒇𝒇𝒇𝒇𝒘𝒘𝒆𝒆𝒆𝒆𝒘𝒘𝑣𝑣 ∈ 𝐷𝐷𝑠𝑠𝑖𝑖𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠𝑑𝑑𝑠𝑠𝑖𝑖[𝑠𝑠]𝒅𝒅𝒇𝒇

𝒘𝒘𝒇𝒇¡ 𝐷𝐷𝑠𝑠𝑠𝑠𝑑𝑑[𝑣𝑣]𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑣𝑣] > 𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] + 𝑓𝑓𝑠𝑠𝑑𝑑𝑔𝑔𝐷𝐷ℎ(𝑠𝑠, 𝑣𝑣)𝒅𝒅𝒇𝒇𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑣𝑣] = 𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] + 𝑓𝑓𝑠𝑠𝑑𝑑𝑔𝑔𝐷𝐷ℎ(𝑠𝑠, 𝑣𝑣)𝑔𝑔𝐷𝐷𝐷𝐷𝑠𝑠𝑑𝑑𝐷𝐷_𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠[𝑣𝑣] = 𝑠𝑠𝐷𝐷𝑑𝑑𝐷𝐷𝑠𝑠𝐷𝐷𝐷𝐷(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠, (𝑣𝑣, 𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑣𝑣]))

C. Prim Algorithm

Prim's algorithm, given a related graph, not directed and weighted, finds a minimal expansion tree. That is, it can find a subset of the edges that form a tree that includes all the vertices of the initial graph, where the total weight of the edges of the tree is the minimum possible (Jungnickel, 1999).

Given a set of nodes N, a set of edges E, and a cost function p, our network (graph) G is defined. To apply the Prim algorithm and solve the problem of the minimum expansion tree, all costs associated with the edges mustn't be negative. It is an algorithm that continuously increases the size of a tree, starting with an initial node, chosen at random, to which nodes are added whose distance successively to the previous ones is minimal. In each step, we will consider the edges that contain nodes that already belong to the tree. The minimum cost expansion tree is entirely constructed when there are no more nodes left to add (Jungnickel, 1999).

𝑃𝑃𝐷𝐷𝐷𝐷𝑖𝑖(𝑔𝑔𝐷𝐷𝐷𝐷𝑔𝑔ℎ𝐺𝐺(𝑠𝑠, 𝑖𝑖, 𝑔𝑔))𝒇𝒇𝒇𝒇𝒇𝒇𝒘𝒘𝒆𝒆𝒘𝒘𝒇𝒇𝒆𝒆𝑠𝑠𝐷𝐷𝑑𝑑𝑠𝑠𝒅𝒅𝒇𝒇𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] = 𝐼𝐼𝑠𝑠𝐼𝐼𝐼𝐼𝑠𝑠𝐼𝐼𝐼𝐼𝐼𝐼𝑔𝑔𝐷𝐷𝐷𝐷𝑠𝑠𝑑𝑑𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠[𝑠𝑠] = 𝑠𝑠𝑁𝑁𝑁𝑁𝑁𝑁𝐼𝐼𝑑𝑑𝐷𝐷𝑠𝑠𝐷𝐷𝐷𝐷(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠,< 𝑠𝑠, 𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] >)

𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] = 0𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘! 𝐷𝐷𝐷𝐷𝑖𝑖𝑖𝑖𝑔𝑔𝐷𝐷𝑖𝑖(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝒅𝒅𝒇𝒇𝑠𝑠 = 𝑠𝑠𝑒𝑒𝐷𝐷𝐷𝐷𝐷𝐷𝑠𝑠𝐷𝐷𝑒𝑒í𝑑𝑑𝐷𝐷𝑖𝑖𝑠𝑠𝑖𝑖(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝒇𝒇𝒇𝒇𝒇𝒇𝒘𝒘𝒆𝒆𝒆𝒆𝒘𝒘𝑣𝑣𝐷𝐷𝑠𝑠𝐷𝐷𝐷𝐷𝑠𝑠𝑠𝑠𝑑𝑑𝐷𝐷𝐷𝐷𝑠𝑠′𝑠𝑠′𝒅𝒅𝒇𝒇

C. Prim Algorithm

Prim’s algorithm, given a related graph, not directed and weighted, finds a minimal expansion tree. That

is, it can find a subset of the edges that form a tree that includes all the vertices of the initial graph, where

the total weight of the edges of the tree is the minimum possible (Jungnickel, 1999).

Given a set of nodes N, a set of edges E, and a cost function p, our network (graph) G is defined. To apply

the Prim algorithm and solve the problem of the minimum expansion tree, all costs associated with the

edges mustn’t be negative. It is an algorithm that continuously increases the size of a tree, starting with

an initial node, chosen at random, to which nodes are added whose distance successively to the previous

ones is minimal. In each step, we will consider the edges that contain nodes that already belong to the

tree. The minimum cost expansion tree is entirely constructed when there are no more nodes left to add

(Jungnickel, 1999).




8

B. Dijkstra's algorithm

The Dijkstra algorithm, also called the minimum path algorithm, is a model that is classified within the search algorithms. Its objective is to determine the shortest route, from the origin node to any node in the network. Its methodology is based on iterations, so that, in practice, its development becomes difficult as the size of the network increases, leaving it at a clear disadvantage, compared to optimization methods based on mathematical programming (Jungnickel, 1999).

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷(𝑔𝑔𝐷𝐷𝐷𝐷𝑔𝑔ℎ𝐺𝐺, 𝐷𝐷𝑠𝑠𝑠𝑠𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝐷𝐷)𝒇𝒇𝒇𝒇𝒇𝒇𝑠𝑠 ∈ 𝑉𝑉[𝐺𝐺]𝒅𝒅𝒇𝒇𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] = 𝐼𝐼𝑠𝑠𝐼𝐼𝐼𝐼𝑠𝑠𝐼𝐼𝐼𝐼𝐼𝐼𝑔𝑔𝐷𝐷𝐷𝐷𝑠𝑠𝑑𝑑𝐷𝐷_𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠[𝑠𝑠] = 𝑠𝑠𝑁𝑁𝑁𝑁𝑁𝑁𝐷𝐷𝑠𝑠𝑠𝑠𝑑𝑑[𝑠𝑠] = 𝑓𝑓𝐷𝐷𝑓𝑓𝐷𝐷𝑠𝑠

𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝐷𝐷] = 0𝐷𝐷𝑑𝑑𝐷𝐷𝑠𝑠𝐷𝐷𝐷𝐷(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠, (𝐷𝐷, 𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝐷𝐷]))𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘! 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑖𝑖𝑖𝑖𝑔𝑔𝐷𝐷𝑖𝑖(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝒅𝒅𝒇𝒇𝑠𝑠 = 𝑠𝑠𝑒𝑒𝐷𝐷𝐷𝐷𝐷𝐷𝑠𝑠𝐷𝐷_𝑖𝑖í𝑑𝑑𝐷𝐷𝑖𝑖𝑠𝑠𝑖𝑖(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝐷𝐷𝑠𝑠𝑑𝑑[𝑠𝑠] = 𝐷𝐷𝐷𝐷𝑠𝑠𝑠𝑠𝒇𝒇𝒇𝒇𝒇𝒇𝒘𝒘𝒆𝒆𝒆𝒆𝒘𝒘𝑣𝑣 ∈ 𝐷𝐷𝑠𝑠𝑖𝑖𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠𝑑𝑑𝑠𝑠𝑖𝑖[𝑠𝑠]𝒅𝒅𝒇𝒇

𝒘𝒘𝒇𝒇¡ 𝐷𝐷𝑠𝑠𝑠𝑠𝑑𝑑[𝑣𝑣]𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑣𝑣] > 𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] + 𝑓𝑓𝑠𝑠𝑑𝑑𝑔𝑔𝐷𝐷ℎ(𝑠𝑠, 𝑣𝑣)𝒅𝒅𝒇𝒇𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑣𝑣] = 𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] + 𝑓𝑓𝑠𝑠𝑑𝑑𝑔𝑔𝐷𝐷ℎ(𝑠𝑠, 𝑣𝑣)𝑔𝑔𝐷𝐷𝐷𝐷𝑠𝑠𝑑𝑑𝐷𝐷_𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠[𝑣𝑣] = 𝑠𝑠𝐷𝐷𝑑𝑑𝐷𝐷𝑠𝑠𝐷𝐷𝐷𝐷(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠, (𝑣𝑣, 𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑣𝑣]))

C. Prim Algorithm

Prim's algorithm, given a related graph, not directed and weighted, finds a minimal expansion tree. That is, it can find a subset of the edges that form a tree that includes all the vertices of the initial graph, where the total weight of the edges of the tree is the minimum possible (Jungnickel, 1999).

Given a set of nodes N, a set of edges E, and a cost function p, our network (graph) G is defined. To apply the Prim algorithm and solve the problem of the minimum expansion tree, all costs associated with the edges mustn't be negative. It is an algorithm that continuously increases the size of a tree, starting with an initial node, chosen at random, to which nodes are added whose distance successively to the previous ones is minimal. In each step, we will consider the edges that contain nodes that already belong to the tree. The minimum cost expansion tree is entirely constructed when there are no more nodes left to add (Jungnickel, 1999).

𝑃𝑃𝐷𝐷𝐷𝐷𝑖𝑖(𝑔𝑔𝐷𝐷𝐷𝐷𝑔𝑔ℎ𝐺𝐺(𝑠𝑠, 𝑖𝑖, 𝑔𝑔))𝒇𝒇𝒇𝒇𝒇𝒇𝒘𝒘𝒆𝒆𝒘𝒘𝒇𝒇𝒆𝒆𝑠𝑠𝐷𝐷𝑑𝑑𝑠𝑠𝒅𝒅𝒇𝒇𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] = 𝐼𝐼𝑠𝑠𝐼𝐼𝐼𝐼𝑠𝑠𝐼𝐼𝐼𝐼𝐼𝐼𝑔𝑔𝐷𝐷𝐷𝐷𝑠𝑠𝑑𝑑𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠[𝑠𝑠] = 𝑠𝑠𝑁𝑁𝑁𝑁𝑁𝑁𝐼𝐼𝑑𝑑𝐷𝐷𝑠𝑠𝐷𝐷𝐷𝐷(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠,< 𝑠𝑠, 𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] >)

𝑠𝑠𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝑑𝑑𝑠𝑠𝑠𝑠[𝑠𝑠] = 0𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘𝒘! 𝐷𝐷𝐷𝐷𝑖𝑖𝑖𝑖𝑔𝑔𝐷𝐷𝑖𝑖(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝒅𝒅𝒇𝒇𝑠𝑠 = 𝑠𝑠𝑒𝑒𝐷𝐷𝐷𝐷𝐷𝐷𝑠𝑠𝐷𝐷𝑒𝑒í𝑑𝑑𝐷𝐷𝑖𝑖𝑠𝑠𝑖𝑖(𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝒇𝒇𝒇𝒇𝒇𝒇𝒘𝒘𝒆𝒆𝒆𝒆𝒘𝒘𝑣𝑣𝐷𝐷𝑠𝑠𝐷𝐷𝐷𝐷𝑠𝑠𝑠𝑠𝑑𝑑𝐷𝐷𝐷𝐷𝑠𝑠′𝑠𝑠′𝒅𝒅𝒇𝒇

9

𝒊𝒊𝒊𝒊((𝑣𝑣 ∈ 𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞)&&(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑞𝑞[𝑣𝑣] > 𝑝𝑝(𝑞𝑞, 𝑣𝑣))𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝑝𝑝𝑑𝑑𝑝𝑝𝑞𝑞𝑑𝑑𝑑𝑑𝑝𝑝𝑝𝑝𝑑𝑑𝑞𝑞[𝑣𝑣] = 𝑞𝑞𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑞𝑞[𝑣𝑣] = 𝑝𝑝(𝑞𝑞, 𝑣𝑣)

𝑈𝑈𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑞𝑞(𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞𝑞,< 𝑣𝑣, 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑞𝑞[𝑣𝑣] >)[10]

3.3. Route size indicator

The size of the route is defined as the sum of the weights of all the edges of the graph that the algorithm has traveled.

For the graph in above Figure 2:

A. Routes that each algorithm has followed • Naive Algorithm: 0 → 1 → 2 • Dijkstra algorithm: 0 → 2 • Prim Algorithm: 0 → 2 B. Route size obtained with each algorithm • Naive Algorithm: 1 + 20 = 21 • Dijkstra Algorithm: 5 • Prim Algorithm: 5

Both the Dijkstra and Prim algorithms have obtained an optimal route size, with a value of 5.

3.4. Algorithmic efficiency complexity indicator

Big O notation is used to asymptotically limit the growth of a runtime that is within constant factors above and below. Sometimes we want to limit only above. It is convenient to have a form of asymptotic notation that means "the execution time grows at most by this much, but it can grow more slowly." We use the "big O" notation just for these occasions. If execution time is O (f (n)), then for a sufficiently large n, the execution time is at most k * f (n) for some constant k (Magzhan & Jani, 2013).

A. The efficiency of the naive algorithm (approximate) 𝑂𝑂(𝑙𝑙𝑙𝑙𝑙𝑙|𝑉𝑉|) = 𝑂𝑂(|𝑙𝑙𝑙𝑙𝑙𝑙|𝑉𝑉|)

B. Dijkstra algorithm efficiency 𝑂𝑂((|𝐴𝐴| + |𝑉𝑉|)𝑙𝑙𝑙𝑙𝑙𝑙|𝑉𝑉|) = 𝑂𝑂(|𝐴𝐴|𝑙𝑙𝑙𝑙𝑙𝑙|𝑉𝑉|)

C. Prim algorithm efficiency 𝑂𝑂(𝑛𝑛!)

To observe results quantitatively, the table 1 will show the data obtained from a previous study (Magzhan & Jani, 2013), regarding Dijkstra's algorithm and others; subsequently, the values will be extrapolated to the cases of the naive algorithm, of

3.3. ROUTE SIZE INDICATOR

The size of the route is defined as the sum of the weights of all the edges of the graph that the algorithm

has traveled.

For the graph in above Figure 2:

A. Routes that each algorithm has followed

• Naive Algorithm: 0 → 1 → 2

• Dijkstra algorithm: 0 → 2

• Prim Algorithm: 0 → 2

B. Route size obtained with each algorithm

• Naive Algorithm: 1 + 20 = 21

• Dijkstra Algorithm: 5

• Prim Algorithm: 5

Both the Dijkstra and Prim algorithms have obtained an optimal route size, with a value of 5.




3.4. ALGORITHMIC EFFICIENCY COMPLEXITY INDICATOR

Big O notation is used to asymptotically limit the growth of a runtime that is within constant factors

above and below. Sometimes we want to limit only above. It is convenient to have a form of asymptotic

notation that means “the execution time grows at most by this much, but it can grow more slowly.” We

use the “big O” notation just for these occasions. If execution time is O (f (n)), then for a sufficiently large

n, the execution time is at most k * f (n) for some constant k (Magzhan & Jani, 2013).

A. The efficiency of the naive algorithm (approximate)

0(log|V|)=(|log|V|)

B. Dijkstra algorithm efficiency

0((|A|+|V|)log|V|)=0(|A|log|V|)

C. Prim algorithm efficiency

0(n2)

To observe results quantitatively, the table 1 will show the data obtained from a previous study (Magzhan

& Jani, 2013), regarding Dijkstra’s algorithm and others; subsequently, the values will be extrapolated to

the cases of the naive algorithm, of complexity O (| log | V |), and Prim, of complexity O (n2), using

the theoretical complexity of each of these algorithms.

Table 1. Results of performance time of the algorithms for different graphs (ms).

Nodes Edges Maximum edge cost

Dijkstra with d-heap O (AlogdN)

(ms)

PrimO (n2)(ms)

NaiveO (|log|V|)

(ms)

100 100 10 0.14 0.12 0.0014

500 100 10 3.77 3.23 0.0377

1000 100 10 11.02 10.34 0.0110

3000 100 10 100.00 90.00 0.0100

Note: It is adapted from Magzhan and Jani (2013), the development of a library for minimum roads, applied to a data protection problem.




Because the Naive algorithm does not consider the comparison to obtain the shortest distance with the

adjacent nodes, unlike the others, it is the most efficient with a relative efficiency of O (| log | V |).

Initially, it was enough to analyze one of the indicators to choose the algorithm indicated for the minimum

route problem, then there was an alleged contradiction between the results of algorithmic efficiency and

route size, however, through the problem of backpack it was discovered that a relationship between

these indicators could be established and we can also accommodate it to different scenarios through the

parameters p and q.

4. INDICATOR OF ALGORITHMIC EFFICIENCY RATIO IN ROUTE SIZE OPTIMIZATIONWhen solving the problem of the selection of route optimizing algorithms, if the algorithmic efficiency

indicator is the guide, we will obtain that the recommended algorithm is naive, but this algorithm is, in

turn, one of the longest routes. Despite being the most efficient, it is not the right algorithm to solve these

types of problems. And by following the criterion of the size of the route, we are not certain that we have

the optimal algorithmic efficiency as Formula (1).

10

complexity O (| log | V |), and Prim, of complexity O (n ^ 2), using the theoretical complexity of each of these algorithms.

Table 1. Results of performance time of the algorithms for different graphs (ms). Nodes Edges Maximum

edge cost Dijkstra with

d-heap 𝑂𝑂(𝐴𝐴𝑙𝑙𝑙𝑙𝑙𝑙!𝑁𝑁) (ms)

Prim 𝑂𝑂(𝑛𝑛")

(ms)

Naive 𝑂𝑂(|𝑙𝑙𝑙𝑙𝑙𝑙|𝑉𝑉|)

(ms) 100 100 10 0.14 0.12 0.0014 500 100 10 3.77 3.23 0.0377

1000 100 10 11.02 10.34 0.0110 3000 100 10 100.00 90.00 0.0100

Note: It is adapted from Magzhan and Jani (2013), the development of a library for minimum roads, applied to a data protection problem.

Because the Naive algorithm does not consider the comparison to obtain the shortest distance with the adjacent nodes, unlike the others, it is the most efficient with a relative efficiency of O (| log | V |).

Initially, it was enough to analyze one of the indicators to choose the algorithm indicated for the minimum route problem, then there was an alleged contradiction between the results of algorithmic efficiency and route size, however, through the problem of backpack it was discovered that a relationship between these indicators could be established and we can also accommodate it to different scenarios through the parameters p and q.

4. Indicator of algorithmic efficiency ratio in route size optimization When solving the problem of the selection of route optimizing algorithms, if the algorithmic efficiency indicator is the guide, we will obtain that the recommended algorithm is naive, but this algorithm is, in turn, one of the longest routes. Despite being the most efficient, it is not the right algorithm to solve these types of problems. And by following the criterion of the size of the route, we are not certain that we have the optimal algorithmic efficiency as Formula (1).

"

($%&'()*+,)-/00)-)/1-2)#∗(('5*/6)7/)$ (1)

In the Knapsack problem, we saw how the apparent contradiction of indicators was solved; similarly, for the problem of selecting algorithms in route optimizers, it is proposed to establish a relationship between algorithmic efficiency and route size, thus generating a new criterion.

A third indicator is proposed through the relationship of the previous two (1), we have both indicators in the denominator because it seeks to minimize the size of the route

(1)

In the Knapsack problem, we saw how the apparent contradiction of indicators was solved; similarly, for

the problem of selecting algorithms in route optimizers, it is proposed to establish a relationship between

algorithmic efficiency and route size, thus generating a new criterion.

A third indicator is proposed through the relationship of the previous two (1), we have both indicators in

the denominator because it seeks to minimize the size of the route and seeks to minimize complexity thus

optimizing algorithmic efficiency. The parameters p and q serve us to graduate the level of importance

that we will give to each indicator, for example, for a brute force algorithm it is advisable to assign at least




a value of 2 to the parameter p and for our route optimizer the value of the parameter q, not It should

be less than 3 due to the great importance of finding the minimum route in emergencies.

What this new indicator expresses to us is the relationship between efficiency and size of the route;

however, it forces to decide to assign a level of importance to the two previous indicators.

Evaluating which algorithm to choose for the graph with the new indicator:

A. With p = 1 y q = 2, is possible get:

• Naive Algorithm: 1/(0(log|3|) * 212)=0.00475261513

• Dijkstra Algorithm: 1/(0(|3|log|3|) *52)=0.02794537699

• Prim Algorithm: 1/(0(32) * 52)=0.00444444444

The one that gets the highest score in the indicator is the Dijkstra algorithm, which is why it should be

implemented in this specific graph and with these parameters.

B. With p = 5 y q = 1, is possible get:

• Naive Algorithm: 1/(0((log|3|)5) * 21)=1.92591398109

• Dijkstra Algorithm: 1/(0/(|3|log|3|)5 * 5)=0.03328740214341

• Prim Algorithm: 1/(0((32)5) * 5)=0.00000338701

The one that obtains the highest score in the indicator is the naive algorithm, due to the enormous

priority given to efficiency (p = 5), recommended in embedded equipment with minimal processing

capacity, it should be implemented in this specific graph and with these parameters.

C. With p = 1 y q = 4 (recommended for route optimizer in ambulances because q ≥ 3), we get:

• Naive Algorithm: 1/(0(log|3|) * 214)=0.0000107769

• Dijkstra Algorithm: 1/(0(|3|log|3|) * 54)=0.00111781507




• Prim Algorithm: 1/(0(32) * 54)=0.00017777777

The one that gets the highest score in the indicator is the Dijkstra algorithm, followed by the Prim

algorithm, so it should be implemented in this specific graph and with these parameters. These values of

parameters p and q are recommended to optimize ambulance routes in an emergency.

It is essential to understand the power of the p and q parameters in the results since they affect the

algorithmic efficiency and the size of the route in an exponential way.

Figure 5. p =1; q = 2. Figure 6. p =5 ; q = 1. Figure 7. p =1 ; q = 4.

Figures 5, 6, and 7 show the behavior of the indicator with the different parameters assumed by p and q

the X-axis, where the red axis represents the algorithmic efficiency and the green Y-axis the size of the

route.

5. STORAGE OF RESULTS IN A DATASETIt is important to store the p and q values, preferably in tuples, for later analysis and to relates them to

the results of the new indicator obtained and the order of the algorithms obtained to form a consistent

dataset and subsequently worked with AI. With the use of the data sets built, various problems can be

solved, such as the minimum path and data science through machine learning.

Machine learning is a technique to implement artificial intelligence (Mediaroom Solutions, 2019;

Bustamante, Rodriguez, & Esenarro, 2019), without programming millions of rules and decision trees.

Its objective is to allow machines to learn automatically and increasingly autonomously. It consists of




learning algorithms that can be of 4 types: supervised, unsupervised, semi-supervised, and reinforced.

A classic example is music recommendation systems based on preferences and songs that the user likes.

6. DISCUSSION ABOUT INDICATORS AND THE KNAPSACK PROBLEM Apparent there are a contradiction of the indicators algorithmic efficiency and route size, differents

criteria recommend different algorithms, which seems to show an apparent contradiction, a similar

situation is presented in the Fractional Knapsack Problem where:

Given n elements e1, e2, ..., in with weights p1, p2, ..., pn and benefits b1, b2, ..., bn, , and given a Knapsack

capable of holding elements up to a maximum of weight M ((Knapsack capacity), that is, we want to find

the proportions of the n elements x1, x2, ..., xn(0 ≤ xi ≤ 1) that we have to introduce in the Knapsack so

that the sum of the benefits of the chosen elements be maximum. That is, we must find values (x1, x2, ...,

xn) so that the benefit is maximized, taking into account that the sum of weights cannot exceed M (the

maximum capacity of the Knapsack) (Joyanes & Zahonero, 2008).

At the moment we have two options to achieve our goal, the first one would be to order our descending

elements according to the benefit and select them when doing this we would be facing the problem in

which we have an item with a great advantage, but with enormous weight.

As a second solution proposal, we have the order of ascending the elements and selecting them; however,

the criterion does not work properly when finding an element with low weight, but with a small benefit.

Failing the criteria of maximum benefit and minimum weight gives the appearance that both requirements

are in contradiction; however, this is not so because there is the alternative of expressing the third

indicator through the relationship of the previous two: bi/pi. Here in the numerator, the criteria are to

maximize and the denominator to minimize.




A voracious algorithm that solves the problem orders the elements in a decreasing way concerning

their ratio bi/pi and adds objects while they fit (Joyanes & Zahonero, 2008). The precondition for this

algorithm is the vectors of weights and benefits are sorted in descending order according to the bi/pi

ratio.

13

weights cannot exceed M (the maximum capacity of the Knapsack) (Joyanes & Zahonero, 2008).

At the moment we have two options to achieve our goal, the first one would be to order our descending elements according to the benefit and select them when doing this we would be facing the problem in which we have an item with a great advantage, but with enormous weight.

As a second solution proposal, we have the order of ascending the elements and selecting them; however, the criterion does not work properly when finding an element with low weight, but with a small benefit.

Failing the criteria of maximum benefit and minimum weight gives the appearance that both requirements are in contradiction; however, this is not so because there is the alternative of expressing the third indicator through the relationship of the previous two: : 𝑏𝑏) /𝑝𝑝).. Here in the numerator, the criteria are to maximize and the denominator to minimize.

A voracious algorithm that solves the problem orders the elements in a decreasing way concerning their ratio 𝑏𝑏) /𝑝𝑝) and adds objects while they fit (Joyanes & Zahonero, 2008). The precondition for this algorithm is the vectors of weights and benefits are sorted in descending order according to the 𝑏𝑏) /𝑝𝑝) ratio.

𝐾𝐾𝐾𝐾𝐾𝐾𝑝𝑝𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾(𝑟𝑟𝑟𝑟𝐾𝐾𝑟𝑟𝑏𝑏𝑟𝑟𝐾𝐾𝑟𝑟𝑏𝑏, 𝑟𝑟𝑟𝑟𝐾𝐾𝑟𝑟𝑤𝑤𝑟𝑟𝑤𝑤𝑤𝑤ℎ𝑡𝑡, 𝑟𝑟𝑟𝑟𝐾𝐾𝑟𝑟𝐾𝐾𝐾𝐾𝑝𝑝, 𝑟𝑟𝑟𝑟𝐾𝐾𝑟𝑟𝐾𝐾𝑠𝑠𝑟𝑟)𝑭𝑭𝑭𝑭𝑭𝑭𝑤𝑤𝑏𝑏𝑟𝑟𝑠𝑠𝑓𝑓0𝑡𝑡𝑠𝑠𝐾𝐾 − 1𝒅𝒅𝑭𝑭𝐾𝐾𝑠𝑠𝑟𝑟[𝑤𝑤] ← 0.0

𝑟𝑟𝑟𝑟𝐾𝐾𝑡𝑡 ← 𝐾𝐾𝐾𝐾𝑝𝑝𝑤𝑤 ← 0𝑾𝑾𝑾𝑾𝑾𝑾𝑾𝑾𝑾𝑾(𝑤𝑤 ≤ 𝐾𝐾 − 1)𝐾𝐾𝐾𝐾𝑎𝑎(𝑤𝑤𝑟𝑟𝑤𝑤𝑤𝑤ℎ𝑡𝑡[𝑤𝑤] ≤ 𝑟𝑟𝑟𝑟𝐾𝐾𝑡𝑡𝑠𝑠)𝒅𝒅𝑭𝑭𝐾𝐾𝑠𝑠𝑟𝑟[𝑤𝑤] ← 1𝑟𝑟𝑟𝑟𝐾𝐾𝑡𝑡𝑠𝑠 ← 𝑟𝑟𝑟𝑟𝐾𝐾𝑡𝑡𝑠𝑠 − 𝑤𝑤𝑟𝑟𝑤𝑤𝑤𝑤ℎ𝑡𝑡[𝑤𝑤]𝑤𝑤 ← 𝑤𝑤 + 1

𝑰𝑰𝑰𝑰𝑤𝑤 < 𝐾𝐾𝒕𝒕𝑾𝑾𝑾𝑾𝒕𝒕𝐾𝐾𝑠𝑠𝑟𝑟[𝑤𝑤] ← 𝑟𝑟𝑟𝑟𝐾𝐾𝑡𝑡𝑠𝑠/𝑤𝑤𝑟𝑟𝑤𝑤𝑤𝑤ℎ𝑡𝑡[𝑤𝑤][12]

As we can see, the weight and benefits indicators were not confronted; it was only necessary to express the relationship between them and the connections used as a new indicator.

7. Conclusions Initially, it was enough to analyze one of the indicators to choose the algorithm indicated for the minimum route problem, then there was an alleged contradiction between the results of algorithmic efficiency and route size, however, through the of Knapsack problem it was discovered that a relationship between these indicators could be established and we can also accommodate it to different scenarios through the parameters p and q.

As we can see, the weight and benefits indicators were not confronted; it was only necessary to express

the relationship between them and the connections used as a new indicator.

7. CONCLUSIONSInitially, it was enough to analyze one of the indicators to choose the algorithm indicated for the minimum

route problem, then there was an alleged contradiction between the results of algorithmic efficiency and

route size, however, through the of Knapsack problem it was discovered that a relationship between

these indicators could be established and we can also accommodate it to different scenarios through the

parameters p and q.

In the search to contribute with the fruits of this study to the solution of various graph problems later

with the results of the algorithms selected by the indicator (1), we can form consistent data sets that

nourish current and future AI techniques.




ACKNOWLEDGMENTSThis paper would not have been possible to realize without the assistance, support, and patience of the

research group. We would like to thank Dra. Magnolia Rueda for her invaluable advice and unsurpassed

knowledge.

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